Chris Pollett > Students > Yunxuan

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    [First Proposal]

    [Final Step of Shor's and Grover's Algorithms]

    [jQuantum QFT]

    [My Quantum Circuits]

    [Three Models]

    [Threshold of Error Correction]

    [jQuantum Manual]

    [Quantum State and LH]

    [QHC Scientists]

    [LH and Tensor Networks]

    [k-LH is QMA-complete]

    [Deutch Josza Algorithm]

    [Semester Report]

    [Second Proposal]

    [SolveQ Algorithm: 2-SAT]

    [Random-kSAT-Generator: Version 1]

    [Freezing Point Experiment]

    [Random-kQSAT-Generator: Version 2]

    [Random-kQSAT-Generator: Version 3]

    [Antiferromagnetic Heisenberg Model]

    [Ising Model]

    [Random-kQSAT-Generator: Version 4]

    [Random-kQSAT-Generator: Version 5]

    [Random-kQSAT-Generator: Version 6]

    [SolveQ Algorithm: 2-QSAT]


The Types of Quantum Computer Models

Ion Trap
The ion trap is a string of atoms that are controlled by laser. Currently we can couple 20 ions and use them to do computation. First, we cool down the ions to about 20 milliKelvins, and the coupling coefficient J between them will cause the ions to become linked in a tensor product. The spin up and spin down of our 0 and 1 are actually not particle spin; the spin here is an abstract description for two atomic energy levels that become coupled in electric dipole coupling. The a electron flops between these two energy levels by the absorbing or emitting a photon, sometimes two photon transitions are also possible. If we shine two lasers on two ions, we essentially have a two qubit gate. Before we can do computation, we have to be able to flip the qubits, and we use laser to achieve this. To do this pushing the atom down with a beam of light will cause a spin up to transition to spin down, while pushing the atom up with a beam of light will cause a spin down to transition to spin up. The transition dipole moment is calculated to determine whether a transition between two states will happen. If this quantity is non-zero, a transition will occur. The next challenge is we have to be able to make a superposition of states of 0 and 1 so that we have the qubit q=a(0)+b(1) . There are two ways to accomplish this. The first way is by rotating the Bloch vector using microwave. The second way is using two detuned lasers. That is two lasers whose frequency are about the same as the energy gap between atomic energy states representing state 0 and 1. One of the lasers, will give you the 0 state and the other the 1 state. So how do we achieve more complicated gates, such as two qubit gates? The phase gate is already possible because the energy difference in pushing up and pushing down will cause a phase to be multiplied to the bell state with 01 and 10 in superposition. Just in case you are curious, the CNOT gate is already possible too. Needless to say, only a very few selected elements can be used for the ion in ion trap. First of all these ions need to have full inner shells. The ion also needs to be light enough so it is easier to push by laser light. Of course it also needs to have atomic energy levels with long life time and the correct energy gap corresponding to available lasers. It also helps for the ion to have a big nucleus to be able to take advantage of hyperfine structure of nuclear spin. The last challenge is how to measure the qubits. It is possible to distinguish the two states in the lab because only the up state will floresce or give off light, so you can measure using a light detector such as a photomultiplier. Right now the researchers are working at coupling 5 to 20 ions. The next step seems to be shooting for 50 qubits in the up coming year.

Quantum Annealer
Both the Shor and Grover algorithms are for ion-traps. It is interesting that not all algorithms useful for information processing are man-made. In fact here we will discuss an algorithm called the Monte-Carlo algorithm that is used by the quantum annealer model to do computation. Even though this algorithm is not as versatile as the Shor's and Grover's algorithms, it is still a valid method of using quantum mechanics to process information.

The Monte-Carlo algorithm takes advantage of a physical process called quantum annealing rather than a series of quantum gates (which themselves are abstract). Quantum annealing is a method for finding the global minimum of a given objective function. Quantum annealing starts from a superposition of all possible states with equal weights. Then the system evolves following the time-dependent Schrödinger equation. As this occurs, the amplitudes of all candidate states keep changing from the original quantum parallelism of the states, according to the time-dependent strength of the tunneling field, which causes quantum tunneling between states. The tunneling field is a transverse field that adds a perturbation that helps the system escape the local minima in a quantum well it encounters using the tunnel effect that allows us to take advantage of the wave-particle duality in order to traverse through the potential dip or barrier by quantum jumps. If the rate of change of the tunneling field is slow, the system stays close to the ground state, the absolute minimum, of the instantaneous Hamiltonian. When the quantum field is finally switched off, and the system is expected to have reached the ground state of the classical Ising model, the state that corresponds to the solution of our problem.

Josephson Junction
The Josephson effect is the phenomenon of a supercurrenta current that flows indefinitely along without any voltage applied across a device known as a Josephson junction, which consists of two superconductors coupled by a weak link. The weak link can be a thin insulating barrier that can be tunneled through by electrons based on quantum mechanics. Before Josephson's prediction, it was only known that normal (non-superconducting) electrons can flow through an insulating barrier, by means of quantum tunneling. Josephson added to this knowledge by predicting the tunneling of superconducting electrons. The phase and charge are two important variables in Josephson junction. The phase is the difference of the phases of the quantum mechanical wave function in two superconducting electrodes forming a Josephson junction. The charge is important because it related to the potential energy as the potential energy accumulated in a Josephson junction when a supercurrent flows through it. It turns out that the state with phase= 0 is unstable and corresponds to the Joseph energy maximum, while the state phase=Pi correspond to the Joseph energy minimum and is a ground state. One advantage of the Josephson Junction is that it can be made into microchips.